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arXiv · 2212.09203

A local gap theorem for Ricci shrinkers

Pak-Yeung Chan·Zilu Ma·Yongjia Zhang
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Abstract

We prove a local gap theorem for Ricci shrinkers, which states that if the local $\mu$-functional at scale $1$ on a large ball centered at the minimum point of the potential function is close enough to $0$, then the shrinker must be the flat gaussian shrinker. In relation to our result, Yokota [Yo09,Yo12] proved the same result assuming the global $\mu$-functional to be close enough to $0$. Our result shows an aspect of how the local geometry of a shrinker controls the global geometry, which is also discussed in [LW19,LW20,LW21].

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Pak-Yeung Chan, Zilu Ma, Yongjia Zhang. 2022-12-19. A local gap theorem for Ricci shrinkers. https://arxiv.org/abs/2212.09203

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